Linear Algebra (incomplete)

Question 1

Linear Algebra

What is a field?

Answer 1

A field is a set F along with 2 binary operations + x st. (F,+,0), (F{0},x,1) are abelian groups and the distributive law (a+b)c=ac+bc holds

Question 2

Linear Algebra

What is the characteristic of a field F?

Answer 2

The smallest number p st. 1+1+1+,,, ( p times ) = 0
p must be prime
If p doesn’t exist we say the characteristic is 0

Question 3

Linear Algebra

What is a vector space V over F?

Answer 3

A set V along with + st. (V,+,0) is an abelian group. Also equipped with a scalar multiplication FxV->V st
for v,w in V and a,b in F
a(v+w)=av+aw, (a+b)v=av+bv, (ab)v=a(bv), 1v=v

Question 4

Linear Algebra

What is a ring?

Answer 4

A ring is a set R equipped with + and x st (R,+,0) is an abelian group and that x is associative and distributive over +

Question 5

Linear Algebra

What is a ring homomorphism

Answer 5

A map f st f(a+b)=f(a)+f(b) and f(ab)=f(a)f(b)

Question 6

Linear Algebra

What is an ideal of a ring R

Answer 6

A subset I st for all a,b in I and r in R
a-b in I
ra and ar in I

Question 7

Linear Algebra

What is the first isomorphism for rings?

Answer 7

The kernel of a ring homomorphism is an idea and is isomorphic to the image of the homomorphism which is a subring.

Question 8

Linear Algebra

State and prove the division algorithm for polynomials

Answer 8

Let f,g in F[x] where g=/=0. Then there exists q,r in F[x] st.
f=gq+r deg(r)
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Question 9

Prove that f(a)=0

=> (x-a) divides f(x)

Answer 9

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Question 10

Linear Algebra
Prove for a(x),b(x) =/= 0 that there exists s and t such that
a(x)s(x)+b(x)t(x)=gcd(a,b)

Answer 10

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Question 11

Linear Algebra

Prove that every ideal in F[x] is generated by one element. ie I={rf : r in R} (T)

Answer 11

Let f,g be monic polynomials of lowest degree in I
Then f-g in I and deg(f-g) certainly in I as r(x)f(x) in I for all r
RTP I in

Let g in I. Then there exists s, t so fs+gt=c=gcd(f,g). fs+gt in I so c in I. => deg(c) >= deg ( f ) => deg(c)=deg(f) => c=λf => g=μf => g in

Question 12

Linear Algebra
Prove:
For any matrix A there exists an f in F[x] so f(A)=0 (T)

Answer 12

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Question 13

Linear Algebra

Define the minimal polynomial of a matrix A

Answer 13

The minimal polynomial is the unique polynomial of least degree such that mA(A)=0

Question 14

Linear Algebra

Prove if f(A)=0 then mA(x) divides f(x). Also show mA(x) is unique

Answer 14

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Question 15

Linear Algebra
Prove
λ is an eigenvalue of A <=>
λ is a root of the characteristic polynomial <=>
λ is a root of the minimal polynomial

Answer 15

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Question 16

Prove the set of cosets V/U is a vector space

Answer 16

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Question 17

Let E be a basis of U and extend to a basis B of V.

Show v+U where v in B\E is a basis for V/U

Answer 17

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Question 18

Let E be a basis for U and let F be a set of vectors st {v+U:v in F} is a basis for V/U
Prove E union F is a basic for V

Answer 18

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Question 19

Prove R(v+A)=T(v)+B is a well defined linear map of quotients A/U->W/B iff T(A) in B

Answer 19

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Question 20

Prove if T(A) in B then there exists a block matrix decomposition

T = ( T¦A * )
( 0 T bar )

Answer 20

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Question 21

Prove if U is T-invariant then Chi T = Chi T¦U * Chi Tbar

Answer 21

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Question 22

Prove if the characteristic polynomial is a product of linear factors then there exists a basis such that T is upper triangular

Answer 22

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Question 23

Prove if A upper triangular with diagonal entries p1, p2, …, pn then
(A-p1I)(A-p2I)…(A-pnI)=-0

Answer 23

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Question 24

State and prove the Cayley Hamilton theorem

Answer 24

If T:V->V is a linear transformation and V is finite dimensional then ChiT(T)=0 and in particular mT(x) divides ChiT(x)
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Question 25

State and prove the primary decomposition theorem

Answer 25

Let
mT=f1^q1*f2^q2*...*fr^qr where the fi are distinct monic irreducible polynomials.
Put Wi=Ker(fi^qi(T))
Then
V=W1+...+Wr (direct sum)
Wi is T-invariant
mT¦Wi = fi^qi

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Question 26

Prove T is diagonalisable iff mT factors as a product of distinct linear polynomials

Answer 26

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